Unproven claim in Weinberg's textbook

[dEdt]

In his Lectures on Quantum Mechanics, Weinberg makes the following claim about solutions to the Schrodinger equation for a central potential: Suppose $\psi(\mathbf{x},t)$ is an eigenfunction of $H$, $\mathbf{L}^2$, and $L_z$. According to Weinberg, "since $\mathbf{L}^2$ acts only on angles, such a wavefunction must be proportional to a function only of angles, with a coefficient of proportionality $R$ that can depend only on $r$. That is, for all r,
$$\psi(\mathbf{x})=R(r)Y(\theta,\phi)."$$

He does not elaborate further on this, in my view, non-trivial statement. Can someone here provide a proof of his claim?

 

[Nugatory]

He does not elaborate further on this, in my view, non-trivial statement. Can someone here provide a proof of his claim?

I'd be inclined to work on the intuition behind that claim instead of looking for a rigorous proof, because the general idea is useful in so many other problems.

But first, which part of the statement are you asking about? The part about $L^2$ acting only on $\theta$ and $\phi$, or the claim that this implies the $R(r)Y(\theta,\phi)$ form of the solution?

 

[dEdt]

I'm talking about the claim that $\psi=R(r)Y(\theta,\phi)$ given that $\mathbf{L}^2$ acts only on angles.

 

[dextercioby]

This is trivial once you consider the uniparticle Hilbert space for the 'dummy' particle 'moving' aroung the CoM. Its Hilbert space is L^2(R^3) = L^2 (R) X L^2 (R) X L^2 (R) and after performing a unitary transformation of the whole space you have that L^2(R^3) ~ L^((0,infinity),dr) X L^2 (S^2,dOmega).

In the position space the wavefunction of the 'dummy' particle would be then a product of a (sq integrable) function of r times a product of a function of spherical angles.

 

[rezeew]

As a fellow scientist, I understand the importance of critically evaluating claims made in scientific literature. In this case, Weinberg's claim may seem non-trivial and lacking in explanation, but it is actually a well-established result in quantum mechanics.

The Schrodinger equation for a central potential is a well-known problem in quantum mechanics, and the solutions to this equation are given by the eigenfunctions of the Hamiltonian, \mathbf{L}^2, and L_z. These eigenfunctions are often referred to as "spherical harmonics," denoted by Y(\theta,\phi), which are functions only of the angles \theta and \phi. This is because the operators \mathbf{L}^2 and L_z only act on the angular variables, leaving the radial variable r unchanged.

Therefore, the wavefunction \psi(\mathbf{x}) can be written as a product of a radial function R(r) and the spherical harmonic Y(\theta,\phi), as stated by Weinberg. This is a consequence of the fact that the Schrodinger equation is separable in spherical coordinates.

In conclusion, Weinberg's claim is a well-established result in quantum mechanics and does not require further proof. However, if you would like to learn more about the mathematical derivation of this result, I recommend referring to a textbook on quantum mechanics or consulting with a quantum physicist.